This problem, we are given that a functional R is a function of three values for three variables U V and W. And that U V and w are all functions of X and Y. As written in this chart right here. And we are asked to find the RdX and D r d Y. Now in this question to use this, we need to use the chain rule, but it's gonna be a little bit longer than we have in the past. And writing this chart on the left hand side is very, very efficient because it shows us that wherever we see an X here, here and here, we're going to have three X is at the bottom, which means that we have to have three different terms to add together. Because there are three elements of this. There is the you part, there's the V park and there's the W part. Okay. And so if we draw this out showing all of this here, we say that we're gonna have three components. The first one is D r d u multiplied by the U D X. We have the R d v times DVD X and we have D R D W time's D W T X. And this can just be exchanged very quickly for the Y2 because all of them are functions of why as well. And I just kind of want to highlight here that, that we are doing this, we should, you should always see that the, the use kind of try to cancel each other out in a way, right? They kind of cancel each other out. We should always get kind of that, you know that that D R D X form in every single term, it's a good way to kind of check yourself. So we just take the derivatives Now D R D the derivative are with respect to you here. Well, the derivative, the natural log is one over the argument. Multiply by the derivative of the inside. Right? Um so we have one divided by U squared plus B squared plus W squared. Have you squared multiplied by the derivative of the inside with respect to you, which is just two times you? Yeah. And then we have the U D X Which is just one. Now this is a very similar feel. Every single time with the derivative are with respect to be so it would be one overuse grip of speed script was W squared multiplied by this time. It's to be then DVD x in this case is just gonna be too, so not too bad. And lastly here, D r d w is going to just be two W divided by U squared plus B squared plus W squared and multiply by D W D X which is going to be equal to two times Y Yeah. Um there's a really, I guess there's a nice way you might be able to simplify this here if you wanted to. Um it's not necessary that you do because you can pull out the U squared plus b squared W. Skirt. I wouldn't do that. I think at this point you can just start plugging things in. So what you can do is you know that X. Is equal to y. It was a little one. We're trying to find the value at excellent like little one but we have a lot of these that are have values of U. V. And W. We can see that plugging in one to you would actually make this 1-plus 2 which is able to three for a deal for V. We would get here that this is equal to just 2 -1 which is equal to one. N. W. Would be equal to just two because we have two times one times one. So wherever we see you we plug in three wherever she will be one. W. is too. So we get That this is going to be six at the top. On the bottom will be three squared plus one squared plus two squared plus two times one divided by three squared plus one squared plus two squared Times two Times 2 and full plus. And we get two times two. This time have three squared plus one squared plus two squared. And then and then why here is equal to one. So we get two times one and we see that this is six divided by so nine square plus one squared plus four squared sorry nine plus one plus four is equal to 14. This is able to to over 14. Oh sorry, Two times two which is equal to 4/14. And lastly here we have two times you damn shoes, which is 8/14 and we get 18/14. This has this baseball can divide by two, So we can get something like 9/7. Mhm. And this is our answer and this is the exact same answer. Not exact same answer, but the same exact process you would do for D. R. D. Y. But do you argue why this part this part and this part do not change, right? These parts do not change because I'm going to have to take the D. R. D. U. And D R D V D R D W. That's that's possible. That's not changing at all. So we would again get, you know, this year? This year and this here the only things that change our that we have to compute now, D. U. D Dy which is equipped to DVDY which is equal to negative one and D. W. D. Y. Which is equal to two. X. Sound good. And so from there we would again get 6/14, multiplied by two Plus. This one would be to over 14. Sorry to over 14 divided by times negative one plus. In this case this would be 4/14 Times two times a which is just to Mhm. And so we get 12/14 -2 or 14 Plus 8/14. And we get that this is actually 18/14. And how about that? We get that. This is 97 again. And so this problem gives us an idea of how to use a function, take a derivative of a function now with three terms, and this extends very very nicely. So this is just important for you to remember. This extends over and over and over again. As we get more and more and more variables draw out this picture, it's extremely help.